Solve 24/18-x-24/18+x=1

$$\frac { 24 } { 18 - x } - \frac { 24 } { 18 + x } = 1$$

Solve for x

$$x=-54$$

$$x=6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -18,18 since division by zero is not defined. Multiply both sides of the equation by \left(x-18\right)\left(x+18\right), the least common multiple of 18-x,18+x.

$$-\left(18+x\right)\times 24-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

To find the opposite of 18+x, find the opposite of each term.

$$\left(-18-x\right)\times 24-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

Use the distributive property to multiply -18-x by 24.

$$-432-24x-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

Use the distributive property to multiply x-18 by 24.

$$-432-24x-\left(24x-432\right)=\left(x-18\right)\left(x+18\right)$$

To find the opposite of 24x-432, find the opposite of each term.

$$-432-24x-24x+432=\left(x-18\right)\left(x+18\right)$$

Combine -24x and -24x to get -48x.

$$-432-48x+432=\left(x-18\right)\left(x+18\right)$$

Add -432 and 432 to get 0.

$$-48x=\left(x-18\right)\left(x+18\right)$$

Consider \left(x-18\right)\left(x+18\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 18.

$$-48x=x^{2}-324$$

Subtract x^{2} from both sides.

$$-48x-x^{2}=-324$$

Add 324 to both sides.

$$-48x-x^{2}+324=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}-48x+324=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -48 for b, and 324 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\left(-1\right)\times 324}}{2\left(-1\right)}$$

Square -48.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\left(-1\right)\times 324}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-48\right)±\sqrt{2304+4\times 324}}{2\left(-1\right)}$$

Multiply 4 times 324.

$$x=\frac{-\left(-48\right)±\sqrt{2304+1296}}{2\left(-1\right)}$$

Add 2304 to 1296.

$$x=\frac{-\left(-48\right)±\sqrt{3600}}{2\left(-1\right)}$$

Take the square root of 3600.

$$x=\frac{-\left(-48\right)±60}{2\left(-1\right)}$$

The opposite of -48 is 48.

$$x=\frac{48±60}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{48±60}{-2}$$

Now solve the equation x=\frac{48±60}{-2} when ± is plus. Add 48 to 60.

$$x=\frac{108}{-2}$$

Divide 108 by -2.

$$x=-54$$

Now solve the equation x=\frac{48±60}{-2} when ± is minus. Subtract 60 from 48.

$$x=-\frac{12}{-2}$$

Divide -12 by -2.

$$x=6$$

The equation is now solved.

$$x=-54 x=6$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -18,18 since division by zero is not defined. Multiply both sides of the equation by \left(x-18\right)\left(x+18\right), the least common multiple of 18-x,18+x.

$$-\left(18+x\right)\times 24-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

To find the opposite of 18+x, find the opposite of each term.

$$\left(-18-x\right)\times 24-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

Use the distributive property to multiply -18-x by 24.

$$-432-24x-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

Use the distributive property to multiply x-18 by 24.

$$-432-24x-\left(24x-432\right)=\left(x-18\right)\left(x+18\right)$$

To find the opposite of 24x-432, find the opposite of each term.

$$-432-24x-24x+432=\left(x-18\right)\left(x+18\right)$$

Combine -24x and -24x to get -48x.

$$-432-48x+432=\left(x-18\right)\left(x+18\right)$$

Add -432 and 432 to get 0.

$$-48x=\left(x-18\right)\left(x+18\right)$$

Consider \left(x-18\right)\left(x+18\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 18.

$$-48x=x^{2}-324$$

Subtract x^{2} from both sides.

$$-48x-x^{2}=-324$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}-48x=-324$$

Divide both sides by -1.

$$\frac{-x^{2}-48x}{-1}=-\frac{324}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{48}{-1}\right)x=-\frac{324}{-1}$$

Divide -48 by -1.

$$x^{2}+48x=-\frac{324}{-1}$$

Divide -324 by -1.

$$x^{2}+48x=324$$

Divide 48, the coefficient of the x term, by 2 to get 24. Then add the square of 24 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+48x+24^{2}=324+24^{2}$$

Square 24.

$$x^{2}+48x+576=324+576$$

Add 324 to 576.

$$x^{2}+48x+576=900$$

Factor x^{2}+48x+576. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+24\right)^{2}=900$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+24\right)^{2}}=\sqrt{900}$$

Simplify.

$$x+24=30 x+24=-30$$

Subtract 24 from both sides of the equation.

$$x=6 x=-54$$